One mole of an ideal gas undergoes a process such that `P prop (1)/(T)`. The molar heat capacity of this process is 4R.

Options

A. The work done by the gas is `2R Delta T`
B. Degree of freedom of the gas is 4
C. Degree of freedom of the gas is 3
D. `gamma` or `((C_(p))/(C_(V)))` for the gas is `1.5`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between pressure and temperature
Given that \( P \propto \frac{1}{T} \), we can express this relationship as:
\[ P = \frac{k}{T} \]
where \( k \) is a constant.

### Step 2: Use the ideal gas law
From the ideal gas law, we know:
\[ PV = nRT \]
For one mole of gas (\( n = 1 \)):
\[ PV = RT \]
Substituting \( P \) from the previous step into the ideal gas equation gives:
\[ \frac{k}{T} V = RT \]

### Step 3: Rearranging the equation
Rearranging the equation:
\[ kV = RT^2 \]
This implies that \( PV \) is a constant, which we can denote as \( C \):
\[ PV = C \]

### Step 4: Determine the molar heat capacity
We are given that the molar heat capacity \( C \) for this process is \( 4R \). The relationship between heat capacities can be expressed as:
\[ C = C_V + \frac{R}{1 - x} \]
where \( x \) is the exponent in the equation \( PV^x = \text{constant} \). Here, since \( P \propto \frac{1}{T} \), we can identify \( x = \frac{1}{2} \).

### Step 5: Substitute values into the heat capacity equation
We know \( C_V = 4R \):
\[ 4R = C_V + \frac{R}{1 - \frac{1}{2}} \]
This simplifies to:
\[ 4R = C_V + 2R \]
Thus:
\[ C_V = 4R - 2R = 2R \]

### Step 6: Calculate \( \gamma \)
The ratio \( \gamma \) (gamma) is defined as:
\[ \gamma = \frac{C_P}{C_V} \]
We can find \( C_P \) using:
\[ C_P = C_V + R = 2R + R = 3R \]
Thus:
\[ \gamma = \frac{C_P}{C_V} = \frac{3R}{2R} = \frac{3}{2} \]

### Step 7: Calculate the degrees of freedom
The degrees of freedom \( F \) can be calculated using:
\[ F = 2 \cdot \frac{1}{\gamma - 1} \]
Substituting \( \gamma = \frac{3}{2} \):
\[ F = 2 \cdot \frac{1}{\frac{3}{2} - 1} = 2 \cdot \frac{1}{\frac{1}{2}} = 4 \]

### Step 8: Conclusion
The degree of freedom is 4, and the value of \( \gamma \) is \( \frac{3}{2} \).